Sophie Germain Identity

The Sophie Germain Identity states that:

One can prove this identity simply by multiplying out the right side and verifying that it equals the left. To derive the factoring, first completing the square and then factor as a difference of squares:

$\begin{align*}a^4 + 4b^4 & = a^4 + 4a^2b^2 + 4b^4 - 4a^2b^2 \\ & = (a^2 + 2b^2)^2 - (2ab)^2 \\ & = (a^2 + 2b^2 - 2ab) (a^2 + 2b^2 + 2ab)\end{align*}$ (Error compiling LaTeX. Unknown error_msg)

Problems

Introductory

• Is a prime?
• Prove that if then is composite.

Intermediate

• Compute . (1987 AIME, #14)
• Find the largest prime divisor of . (Mock AIME 5 2005-2006 Problems/Problem 5)

See Also

• MacTutor biography of Sophie Germain